{"id":"openai-math/Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six","pdf":"https://github.com/openai/math/blob/main/preprints/Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026/Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026.pdf","url":"https://github.com/openai/math/tree/main/preprints/Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026","tags":["combinatorics","hadamard-matrices","mutually-unbiased-bases"],"score":0,"title":"Exact Fourier certificates for complex Hadamard matrices of order six","domain":"Mathematical physics","family":"266","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.MP","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-24","reactions":{},"formalized":false,"family_title":"Exactly three mutually unbiased bases in dimension six"}