Exact Fourier certificates for complex Hadamard matrices of order six

We prove the Fourier-vanishing conjecture of Matolcsi, Ruzsa, and Weiner: every complex Hadamard matrix of order six outside Tao's cubic equivalence class has a vanishing character sum at every permutation of $(1,1,1,-1,-1,-1)$. We also give an exact certificate excluding seven mutually unbiased bases in ℂ${}^6$; by Weiner's completion theorem, this yields an upper bound of five. Both certificates use only integer and rational arithmetic, and the complete verifier is included.