Proves that real circulant Hadamard matrices exist exactly in orders 1 and 4, resolving the circulant Hadamard conjecture. Together with classical Barker-sequence results, this shows that binary sequences whose nontrivial aperiodic autocorrelations have magnitude at most 1 exist at lengths n > 1 exactly when \(n\in\{2,3,4,5,7,11,13\}\). (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 two independent assessments disagree on how much of the result described above is expressed by the Lean statements. Read the assessments
In The circulant Hadamard conjecture, 23 September 2026
Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
For a positive integer n, a real circulant Hadamard matrix of order n exists if and only if n\in\{1,4\}.
In The circulant Hadamard conjecture, 23 September 2026
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
For an integer n>1, a Barker sequence of length n exists if and only if
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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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.
CirculantHadamardThe paper states
In The circulant Hadamard conjecture, 23 September 2026
For a positive integer n, a real circulant Hadamard matrix of order n exists if and only if n\in\{1,4\}.
The words they share: circulant, exists, hadamard, matrix, order, positive, real.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
CirculantHadamard.lean, the release lists it as a main result.
theorem exists_iff_order_one_or_four (n : ℕ) (hn : 0 < n) : ExistsRealCirculantHadamard n ↔ n = 1 ∨ n = 4 := 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
EvenBarkerThe paper states
In The circulant Hadamard conjecture, 23 September 2026
For an integer n>1, a Barker sequence of length n exists if and only if
The words they share: length, barker, integer, sequence.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
EvenBarker.lean, the release lists it as a supporting statement.
theorem even_length_eq_two_or_four {n : ℕ} (h : Fin n → ℤ) (hnpos : 0 < n) (hn : Even n) (hb : IsBarker h) : n = 2 ∨ n = 4 := 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.
IsBarker Model.lean, line 16IsSign Model.lean, line 9IsCirculant Model.lean, line 14IsSignHadamard Model.lean, line 17ExistsRealCirculantHadamard Model.lean, line 21MathVet and cert-machine disagree on how much of the advertised result the Lean statements express. MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”); cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — DECLARED”).
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: exists_iff_order_one_or_four (n) (hn : 0 < n) : ExistsRealCirculantHadamard n ↔ n = 1 ∨ n = 4 with IsCirculant H := ∃ h, ∀ i j, H i j = h (j - i) and IsSignHadamard H := (∀ i j, IsSign (H i j)) ∧ H * Hᵀ = n • 1: exact statement of the headline. The Barker corollary is only half stated (even_length_eq_two_or_four : Even n → IsBarker h → n = 2 ∨ n = 4); odd lengths and the ‘exist exactly at {2,3,4,5,7,11,13}’ classification are not in Lean (docs: ‘outside this selected additional statement’).
“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 a narrower statement than the advertised result (their term: “NARROWER — DECLARED”).
The gap they name: Odd Barker lengths and existence at 2,3,4,5,7,11,13.
“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
The formalization proves the circulant Hadamard conjecture in exact form: a real circulant Hadamard matrix of positive order $n$ exists exactly when $n=1$ or $n=4$. Explicit witnesses are supplied for both orders, with no restriction on prime factors.
It also proves the even-length part of the Barker-sequence consequence. A positive even-length sign sequence whose nonzero aperiodic autocorrelations have absolute value at most one must have length $2$ or $4$. The paper’s classification of odd Barker lengths is outside this selected additional statement.
Classification of circulant Hadamard orders CirculantHadamard
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: a real circulant Hadamard matrix of positive order n exists exactly when n = 1 or n = 4.
Classification of positive even Barker lengths EvenBarker
Unreviewed AI draft: states a consequence of a paper’s theorem. States the even-length part of the paper’s Barker-sequence corollary: even Barker sequences have length 2 or 4. The classification of odd lengths is outside the 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.
CirculantHadamardTwo 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 |
EvenBarkerTwo 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.
anxkhn, Circulant Hadamard: link the OpenAI Lean proof and update the status (issue 6896), 7 October 2026. Source
Exact release theorem checked natively on macOS with Lean 4.34.1: only the three standard axioms; bridge to the Formal Conjectures definition compiles with the same axioms.
Robert Joseph George, Do the Lean statements in openai/math match their papers?, 7 October 2026. Source
The review reports that Comparator accepted OpenAI’s solution for this challenge, run end to end on Lean 4.34.1 with the default kernel and only the three standard axioms.
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
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: CirculantHadamard, EvenBarkerSuggest 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 circulant Hadamard result and records that the human statement uses the product of the matrix and its transpose in the other order.
mo271, Arxiv/2104.00502/BarkerSequence: barker_conjecture is true (issue 6917), 7 October 2026. Source
The circulant Hadamard manuscript implies every Barker sequence has length at most 13, but the release’s Lean proof (EvenBarker) covers only even lengths.
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 circulant Hadamard conjecture, 23 September 2026
Why this passage is shown
Corollary 1.2 (Barker-sequence lengths)
In The circulant Hadamard conjecture, 23 September 2026
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, CirculantHadamard
Proof-verification report
MathVet, CirculantHadamard
Proof-verification report
cert-machine, EvenBarker
Proof-verification report
MathVet, EvenBarker
Lean statement
CirculantHadamard
Lean statement
EvenBarker