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Exactly three mutually unbiased bases in dimension six
Proves , resolving Zauner's dimension-six mutually unbiased bases conjecture: three such bases exist in ℂ, but four cannot. The exclusion is a complete certified computation under the stated binary64 arithmetic and compiler conditions. An independent companion proves the Matolcsi–Ruzsa–Weiner Fourier-vanishing conjecture for order-six complex Hadamard matrices outside Tao's cubic equivalence class.
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The maximum number of mutually unbiased bases in dimension six
Sep 24, 2026·The-maximum-number-of-mutually-unbiased-bases-in-dimension-six-September-24-2026.pdf
We prove that the maximum number of mutually unbiased orthonormal bases in ℂ is three, resolving Zauner's dimension-six MUB conjecture. The upper bound is computer-assisted: under the stated binary64 arithmetic and compiler conditions, a complete execution of the documented verification pipeline excludes four arbitrary complex bases.
Exact Fourier certificates for complex Hadamard matrices of order six
Sep 24, 2026·Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026.pdf
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 . We also give an exact certificate excluding seven mutually unbiased bases in ℂ; 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.