Using topology to reveal all mutually unbiased bases in dimension six

Mutually unbiased bases (MUBs) describe complementary quantum measurements: certainty in one basis gives no information about the others. In dimension six, triples of MUBs are known, but whether a fourth basis could exist remained unresolved for over three decades. OpenAI recently reported a proof that none exists. Here we use the topology of the space of complex Hadamard matrices, which describe transitions between MUBs, to characterize all sets in dimension six. We prove that every mutually unbiased triple belongs to a two-parameter family or is a single isolated triple built from the Fourier matrix. Since none of them extend to a fourth basis, this also proves that no four MUBs exist. Our proof rests on a new topological result that all order-six complex Hadamard matrices except Tao's form a connected space. Connectedness reduces the classification of triples to finitely many overlapping four-dimensional phase balls. We certify with interval arithmetic that these balls cover the space and for every pair of MUBs represented in a ball, we then determine all third bases that extend it. Dimension six is now the first non-prime-power dimension in which all sets of MUBs are classified.