Triplets of mutually unbiased bases in dimension six

We prove that every triplet of mutually unbiased bases of ℂ⁶ is equivalent, by a unitary map and a relabeling of the bases and of their vectors, up to phases, to a triplet from Zauner's construction. This is Conjecture 1 of Matolcsi, Matszangosz, Varga and Weiner, which we call the Budapest conjecture: up to equivalence, each such triplet lies in exactly one of two cases, the two-parameter family described by Szöllősi or one isolated triplet. The four authors' Conjecture 2, which they regard as the main conjecture of their paper, holds as well. Cárdenes Wuttig and Tindall have posted a different computer-assisted proof of the Budapest conjecture. We show without a computer that a triplet comes from the construction exactly when it has a unitary symmetry of order three up to a scalar that fixes every vector of one basis up to a phase and permutes the vectors of the other two up to phases; we call such a symmetry a clock, as Sun does for the Fourier family. To show that every triplet has a clock, we follow it through a finite cover of the complex Hadamard matrices of order six up to equivalence by certified balls, along an acyclic directed graph, to a ball near the Fourier family, where its clock is certified. In addition, we determine which pairs of bases lie in a triplet, and we recover that there are no four mutually unbiased bases in ℂ⁶. The proof is computer-assisted and assumes stated hypotheses on the computations (Appendix A).