Exactly three mutually unbiased bases in dimension six
OpenAI research collection · 2 manuscripts · Source checked 2026-10-19
OpenAI's result summary
Proves N(6)=3, resolving Zauner's dimension-six mutually unbiased bases conjecture: three such bases exist in ℂ^(6), 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.
Read the manuscripts
Short excerpts below come from OpenAI's abstracts. The full abstract and original PDF preserve the paper's precise scope.
MANUSCRIPT 01
The maximum number of mutually unbiased bases in dimension six
We prove that the maximum number of mutually unbiased orthonormal bases in ℂ^(6) 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…
Full source abstract
We prove that the maximum number of mutually unbiased orthonormal bases in ℂ^(6) 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.
This exact PDF is not listed in the imported formalization catalogue. Family scope notes may contain further information.
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…
Full source abstract
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.
This exact PDF is not listed in the imported formalization catalogue. Family scope notes may contain further information.
New mathematical reasoning could help AI tackle harder scientific and computational problems. How reliably these skills transfer beyond mathematics will determine how much they contribute to general intelligence.