Source authenticated

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

Theorem 1.1: a real circulant Hadamard matrix of order $n\ge1$ exists if and only if $n\in\{1,4\}$ . Section 2 reproves Turyn's restriction $n=4u^2$ with $u$ odd, and the main argument excludes every odd $u>1$ through an alternating product of character values over the primes dividing $u$ . Corollary 1.2 deduces the Barker-sequence classification. It says nothing about the (non-circulant) Hadamard conjecture on existence in every order divisible by 4, nor about complex or group-developed variants.

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

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

Open the source record ↗

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

specialist registry · passed

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture as stated in the manuscript and in Ryser's tradition; it is the full conjecture with no restriction on the order. The proof (group-ring and cyclotomic arguments) was not refereed. lean/formalization.yaml lists a main result for this manuscript (comparator CirculantHadamard, declaration OAI.CirculantHadamard.exists_iff_order_one_or_four). The comparator statement CirculantHadamard.lean was read here: for every $n>0$ , a real $n\times n$ matrix with $\pm1$ entries, $H_{ij}=h(j-i)$ with index arithmetic modulo $n$ , and $HH^T=nI$ exists if and only if $n=1$ or $n=4$ . This states the headline claim. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice. The manuscript lists five earlier claimed complete proofs (Oh-Hashi 2016, Orozco Lopez 2019, Morris 2023, Gallardo 2024, Manjhi-Kumar 2025) without assessing them.

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 (OAI.CirculantHadamard.exists_iff_order_one_or_four)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

Source ↗

Comparator statement: CirculantHadamard.lean

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

Source ↗

Artifacts and verifiers

Lean proof (OAI.CirculantHadamard.exists_iff_order_one_or_four)

lean artifact · pending

Artifact ↗

Comparator statement: CirculantHadamard.lean

lean artifact · pending

Artifact ↗

Compute record

No linked compute attempts recorded.

Lineage and corrections

This event attributed to Unreleased internal OpenAI model

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

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

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

Comparator statement: CirculantHadamard.lean evidence for this event

Lean proof (OAI.CirculantHadamard.exists_iff_order_one_or_four) evidence for this event

Act on this frontier

Verify, challenge, or extend the result.

Help verify Open canonical problem Explore evidence graph