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.

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.