Result 179, Combinatorics

The circulant Hadamard and Barker-sequence conjectures

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∈{2,3,4,5,7,11,13}n\in\{2,3,4,5,7,11,13\}.

Lean formalization Proof

The bigger picture

Why it matters

How long can a string of plus and minus signs remain almost uncorrelated with shifted copies of itself? The manuscript claims an exact limit, linked to a classification of square arrays with perfectly orthogonal cyclically shifted rows.

What changes?

The unreviewed manuscript reports that real circulant Hadamard matrices exist exactly in orders 1 and 4. These are square arrays of plus and minus ones whose rows are cyclic shifts of one row and whose distinct rows have dot product zero. Together with classical results, this gives exactly the Barker-sequence lengths above one: 2, 3, 4, 5, 7, 11 and 13. Such sequences have every nonzero-shift correlation, computed over overlapping entries without wrapping, of magnitude at most one.

What does that help mathematicians do?

The claimed classification would rule out an entire infinite family of candidates, rather than merely exclude lengths checked by computation. In particular, every sequence of plus and minus ones longer than 13 would necessarily have some nonzero shift whose overlapping product sum has magnitude at least two. Researchers could therefore stop seeking longer sequences satisfying the Barker condition and focus on precisely stated relaxations.

Are there practical applications?

The immediate value is foundational: the result would identify exact limits on orthogonality under cyclic shifts and near-cancellation under shifts without wrapping. These are different correlation requirements, and the reported consequence connects their limits. Any practical use would need additional work on constructions with weaker requirements; the supplied account does not describe such constructions or demonstrate performance gains.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

The circulant Hadamard conjecture

September 23, 2026 15 pages Main result formalized in Lean

We prove the circulant Hadamard conjecture: a real circulant Hadamard matrix has order 1 or 4. As a consequence, Barker sequences of length greater than one exist exactly at lengths 2, 3, 4, 5, 7, 11, 13, proving the Barker-sequence conjecture.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:The-circulant-Hadamard-conjecture-September-23-2026,
  author = {{OpenAI}},
  title = {{The circulant Hadamard conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-circulant-Hadamard-conjecture-September-23-2026/paper.pdf}{OAI:The-circulant-Hadamard-conjecture-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/179.md.

The circulant Hadamard and Barker-sequence conjectures

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization proves the circulant Hadamard conjecture in exact form: a real circulant Hadamard matrix of positive order nn exists exactly when n=1n=1 or n=4n=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 22 or 44. The paper's classification of odd Barker lengths is outside this selected additional statement.

Comparator links

Result Comparator statement
Classification of circulant Hadamard orders CirculantHadamard.lean
Classification of positive even Barker lengths EvenBarker.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.