Barker-sequence length conjecture
Status OpenLLM status Open unverified
Top Open Problems unknownunknown Not reported cataloguedOpenOpenStatement
Definitions and mathematical statement
Statement recorded in the supplied source.
Must every {-1,1} sequence of length $n>13$ have a nonzero-shift aperiodic autocorrelation with absolute value greater than one?Short English statement
Must every {-1,1} sequence of length $n>13$ have a nonzero-shift aperiodic autocorrelation with absolute value greater than one?Sources
- S1 Original question source or mathematical context. https://arxiv.org/abs/1501.06035.
- S2 Original question source or mathematical context. https://arxiv.org/abs/1501.06035.
- S3 OpenAI, manuscript citation page. Source of the formulation motivating this local problem. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-circulant-Hadamard-conjecture-September-23-2026/README.md.
Statement licence: CC-BY-4.0 · Source file
Contributions 1
OpenAI claim summary for Math-2598
The Barker corollary classifies every length $n>1$, of either parity, as possible exactly for $2,3,4,5,7,11,13$. Hence its claimed result proves that every sign sequence of length greater than thirteen has a nonzero-shift aperiodic autocorrelation of magnitude greater than one. The linked Lean scope covers only the portions specified there.
Barker-sequence length conjecture
OpenAI's claim
Circulant Hadamard orders and Barker lengths: The manuscript claims that real circulant Hadamard matrices occur only at orders one and four. Its further consequence restricts Barker sequences longer than one to the seven lengths $2,3,4,5,7,11,13$.Mathematical statement of the claimed result
Mathematical claim 1.
A real circulant sign matrix $H\in\{-1,1\}^{n\times n}$ with $HH^{\mathsf T}=nI$ is claimed to exist exactly for $n\in\{1,4\}$. The Barker corollary gives an aperiodic sign sequence $(a_0,\ldots,a_{n-1})$ with $|\sum_{j=0}^{n-k-1}a_ja_{j+k}|\le1$ for every $1\le k<n$, at an integer length $n>1$, exactly when \[n\in\{2,3,4,5,7,11,13\}.\] Thus the longer-length impossibility assertions are proved according to the manuscript; they are not counterexamples to those assertions.Outline of the argument
Circulant Hadamard orders and Barker lengths: The first row is written in an integral cyclic group ring, where orthogonality becomes a scalar norm equation. Cyclotomic coefficient divisibility first reduces the order to $4u^2$ with odd $u$. An alternating product of character evaluations over the prime components is then followed through descent, ruling out $u>1$; standard correlation reductions give the Barker consequence.Formal verification
Family 179 describes Lean coverage for circulant order classification and even Barker lengths; the odd Barker classification is not selected.Circulant Hadamard orders and Barker lengths: The manuscript links CirculantHadamard.lean (declaration OAI.CirculantHadamard.exists_iff_order_one_or_four), EvenBarker.lean.
The documented selection does not cover the entire manuscript assertion relevant here.
This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.
Status caveat
OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.Sources
- OpenAI, The circulant Hadamard conjecture, 2026-09-23; citation key
OAI:The-circulant-Hadamard-conjecture-September-23-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: cor:barker; Statement source: thm:main. - Family 179: Lean scope.
- Comparator CirculantHadamard.json.
- Comparator CirculantHadamard.lean.
- Comparator EvenBarker.json.
- Comparator EvenBarker.lean.
- OpenAI catalogue entry.