Fourier character sums for order-six complex Hadamard matrices
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For order-six complex Hadamard matrices outside Tao’s cubic equivalence class, does the normalized character sum vanish at every permutation of (1,1,1,−1,−1,−1), allowing row and column permutations and arbitrary unit phases?
Why it matters and what counts as progress
Why it matters: It asks whether a Fourier constraint known for special families holds throughout the full non-Tao class.
What counts as progress: Prove character-sum vanishing for all non-Tao order-six Hadamard matrices, rather than only parametrized families.
How the problem developed
- 2014 · Partial result [1]
Andrew Maxwell and Stephen Brierley Proved the conjectured Fourier constraint for matrices in the three-parameter Karlsson family, a restricted family of order-six Hadamard matrices.
- 24 Sept 2026 · Reported claim [2]
OpenAI Claimed Fourier-character vanishing for every order-six complex Hadamard matrix outside Tao’s cubic equivalence class, at all permutations of (1,1,1,−1,−1,−1).
AI contribution
Exact Fourier certificates for complex Hadamard matrices of order six
For every order-six complex Hadamard matrix outside Tao’s cubic equivalence class, the normalized character sum vanishes at every permutation of (1,1,1,−1,−1,−1), allowing row and column permutations and arbitrary unit phases.
Sources
- Andrew Maxwell and Stephen Brierley · On properties of Karlsson Hadamards and sets of Mutually Unbiased Bases in dimension six
- OpenAI, exact Fourier certificates for order-six Hadamard matrices
- Exact Fourier certificates for complex Hadamard matrices of order six
- OpenAI, exact Fourier certificate verification
- OpenAI, formalization scope notes
- OpenAI, Lean theorem for order-six Fourier vanishing
Website timeline
Publication dates and editorial changes.
OpenAI reports Fourier vanishing for order-six Hadamard matrices