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The Barker-sequence conjecture: no Barker sequence of length greater than 13

Corollary 1.2: for $n>1$ , a Barker sequence of length $n$ exists if and only if $n\in\{2,3,4,5,7,11,13\}$ . The new input is the even case, via the circulant Hadamard theorem; the odd case is the classical Turyn-Storer theorem. Section 5 records examples at every listed length. Nothing is claimed about nearly-Barker sequences or the asymptotic merit-factor problem.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The Barker-sequence conjecture: no Barker sequence of length greater than 13 · Barker-sequence conjecture

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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Attribution

VibeMathed
registry · event recorded by

Unreleased internal OpenAI model
model · ai model contributor · OpenAI

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: The Barker-sequence conjecture: no Barker sequence of length greater than 13

specialist registry · passed

No independent mathematician has checked this yet. Checked here: Corollary 1.2 and Section 5 were read; the even case reduces to Theorem 1.1 by the classical periodic-autocorrelation argument, and the odd case is quoted from Turyn-Storer and Schmidt-Willms. The challenge lean/ComparatorChallenges/EvenBarker.json (solution module OAI.LinearAlgebra.Barker.Main, present at the pinned commit) is not in the formalization catalogue; its statement EvenBarker.lean was read here: a $\pm1$ integer sequence of positive even length $n$ with all aperiodic autocorrelations at shifts $0<k<n$ of absolute value at most 1 has $n=2$ or $n=4$ . This covers the even half, which is the new part; the odd-length classification is not formalized. Not rebuilt here.

Source ↗

The circulant Hadamard conjecture: real circulant Hadamard matrices exist only in orders 1 and 4

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

Source ↗

Lean proof, even lengths (OAI.CirculantHadamard.Barker.even_length_eq_two_or_four)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

Source ↗

Comparator statement: EvenBarker.lean

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

Source ↗

Artifacts and verifiers

Lean proof, even lengths (OAI.CirculantHadamard.Barker.even_length_eq_two_or_four)

lean artifact · pending

Artifact ↗

Comparator statement: EvenBarker.lean

lean artifact · pending

Artifact ↗

Compute record

No linked compute attempts recorded.

Lineage and corrections

This event attributed to Unreleased internal OpenAI model

VibeMathed record: The Barker-sequence conjecture: no Barker sequence of length greater than 13 evidence for this event

Comparator statement: EvenBarker.lean evidence for this event

Lean proof, even lengths (OAI.CirculantHadamard.Barker.even_length_eq_two_or_four) evidence for this event

The circulant Hadamard conjecture: real circulant Hadamard matrices exist only in orders 1 and 4 evidence for this event

The Barker-sequence conjecture: no Barker sequence of length greater than 13 parent of this event

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