Exactly three mutually unbiased bases in dimension six. Proves $N(6)=3$, resolving Zauner's dimension-six mutually unbiased bases conjecture: three such bases exist in ℂ6, 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.
released 2026-09-24 | 2 theorems · 43 lemmas · 57 proofs · 33,438 words |
PLAY LEVEL 1 »(pdf)
We prove that the maximum number of mutually unbiased orthonormal bases in ℂ6 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.
released 2026-09-24 | 2 theorems · 7 lemmas · 14 proofs · 11,927 words |
PLAY LEVEL 2 »(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 $(1,1,1,-1,-1,-1)$. We also give an exact certificate excluding seven mutually unbiased bases in ℂ6; 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.