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. (Lean)
Manuscripts (2)
The maximum number of mutually unbiased bases in dimension six
September 24, 2026 · source · cite as \cite{OAI:The-maximum-number-of-mutually-unbiased-bases-in-dimension-six-September-24-2026}
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.
Exact Fourier certificates for complex Hadamard matrices of order six
September 24, 2026 · source · cite as \cite{OAI:Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026}
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.
Result 266 · Mathematical physics. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md