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The Matolcsi-Ruzsa-Weiner Fourier-vanishing conjecture for complex Hadamard matrices of order six

Theorem 1.1: every complex Hadamard matrix of order six not equivalent to Tao's cubic matrix has $g_H(\pi\alpha)=0$ for every permutation $\pi$ of $\alpha=(1,1,1,-1,-1,-1)$ . Theorem 1.2, by a second exact certificate and Weiner's completion theorem, excludes seven mutually unbiased bases in $\mathbb C^6$ , giving $N(6)\le5$ . It does not prove the further Matolcsi-Matszangosz-Varga-Weiner conditions for MUB triplets or the bound $N(6)=3$ , which is the companion's computer-assisted claim.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 24, 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 Matolcsi-Ruzsa-Weiner Fourier-vanishing conjecture for complex Hadamard matrices of order six · Matolcsi-Ruzsa-Weiner Fourier vanishing

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 Matolcsi-Ruzsa-Weiner Fourier-vanishing conjecture for complex Hadamard matrices of order six

specialist registry · passed

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the manuscript was read against MRW Conjecture 2.3; the manuscript notes its normalization divides the conjecture's sums by six, which does not affect vanishing. The verifier was not re-run here. Lean: lean/ComparatorChallenges/MUBSix.json exists with solution_module OAI.Analysis.MutuallyUnbiased.Main, whose file exists at the pinned commit; this challenge is not in lean/formalization.yaml. The statement MUBSix.lean was read here: its first conjunct says that for every 6x6 complex matrix with unimodular entries and $H^*H=6I$ that is not equivalent (row and column permutations and unit phases) to the cube-root matrix with Tao's exponent pattern, the normalized character sum vanishes at every permutation of $(1,1,1,-1,-1,-1)$ . This states the headline; the second conjunct is the bound of five bases. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice.

Source ↗

The Matolcsi-Ruzsa-Weiner Fourier-vanishing conjecture for complex Hadamard matrices of order six

witness · pending

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

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Lean proof (OAI.MUB6.fourier_and_family_bound)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

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Comparator statement: MUBSix.lean

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

Source ↗

Related in family: The maximum number of mutually unbiased bases in dimension six

paper · pending

paper link supplied by the VibeMathed registry.

Source ↗

Artifacts and verifiers

Lean proof (OAI.MUB6.fourier_and_family_bound)

lean artifact · pending

Artifact ↗

Comparator statement: MUBSix.lean

lean artifact · pending

Artifact ↗

Compute record

No linked compute attempts recorded.

Lineage and corrections

The Matolcsi-Ruzsa-Weiner Fourier-vanishing conjecture for complex Hadamard matrices of order six evidence for this event

VibeMathed record: The Matolcsi-Ruzsa-Weiner Fourier-vanishing conjecture for complex Hadamard matrices of order six evidence for this event

This event attributed to Unreleased internal OpenAI model

The Matolcsi-Ruzsa-Weiner Fourier-vanishing conjecture for complex Hadamard matrices of order six parent of this event

Related in family: The maximum number of mutually unbiased bases in dimension six evidence for this event

Comparator statement: MUBSix.lean evidence for this event

Lean proof (OAI.MUB6.fourier_and_family_bound) evidence for this event

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