OpenAI · 179-01 · 2026-09-23 · Combinatorics
The circulant Hadamard conjecture
preprints/The-circulant-Hadamard-conjecture-September-23-2026
Family 179. The circulant Hadamard conjecture.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Proves that real circulant Hadamard matrices exist exactly in orders and , resolving the circulant Hadamard conjecture. Together with classical Barker-sequence results, this shows that binary sequences whose nontrivial aperiodic autocorrelations have magnitude at most exist at lengths exactly when .
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.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 179: The circulant Hadamard and Barker-sequence conjectures. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.