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M(6) = 3: An Unconditional Proof

Pompetzki, Christopher

Published Dec 12, 2025• 0 citations
Publisher

Abstract

This paper presents an unconditional proof that the maximum number of mutually unbiased bases in dimension six satisfies M(6) = 3. The argument avoids classification assumptions and proceeds via explicit structural obstructions.
DOI
10.5281/zenodo.17911555

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We prove that the maximum number of mutually unbiased bases in complex dimension sixis three. The proof is structural and does not rely on exhaustive classification, numerical search,or probabilistic arguments. The key ingredients are: (1) a Laurent–polynomial obstructionblocking the Haagerup family of complex Hadamard matrices, (2) an analogous obstructionblocking the 2-circulant family, and (3) a structural Bridge Lemma showing that any complexHadamard matrix arising as a transition matrix in a MUB triplet must exhibit an opposite-pairphenomenon in its row-ratio multisets. Combined with the recognition theorem of Matszangoszand Sz¨oll˝osi, this forces all MUB-triplet transition matrices in dimension six into families alreadyobstructed, yielding a contradiction to the existence of a fourth mutually unbiased basis.