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Exact Fourier certificates for complex Hadamard matrices of order six
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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 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.
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Notes
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Hao-Yu Sun, Mutually unbiased bases: common states, completion, and extension obstructions (arXiv:2610.12180, 2026-10-08). Cites this preprint alongside its companion, states "Neither preprint is a premise of the proofs in this article," and proves an unextendibility result for a specific dimension-six MUB triple. [family-level]
Comments and discussion
- mo271 and commenters, OpenQuantumProblems/13:
mutuallyUnbiasedBases_dim6has answer 3 (google-deepmind/formal-conjectures issue #6918, opened 2026-10-07). A comment of 2026-10-11 reports a re-implementation of this paper's μ(6) ≤ 5 certificate from the LaTeX in exact rational arithmetic, with matching table sizes and ranks; the comment says the Lean proof was not built. [family-level] [AI-generated per source]
Formalization
- OpenAI math release, Lean scope note for family 266 (lean/docs/266.md, 2026). The release's Lean development proves the Fourier vanishing statement for order-six Hadamard matrices not equivalent to Tao's matrix, a family bound of five, and a supporting cube-fiber cancellation lemma (
MUBSix.lean,HadamardCubeFiber.lean), distinguishing these from the companion paper's exclusion of four arbitrary bases. [family-level] - Google DeepMind formal-conjectures, FormalConjectures/OpenQuantumProblems/13.lean (GitHub, accessed 2026-10-21). Marks the variant
mutuallyUnbiasedBases_dim6_bounds.variants.upper_five(μ(6) ≤ 5) research solved with aformal_prooflink to the release's Lean proof and cites this preprint, while leaving the dimension-six maximum open withsorry. [cites earlier version] [family-level]
Possibly relevant literature (found by the agent, not checked)
- Anonymous, No Szöllősi matrix lies in a quadruplet of mutually unbiased bases in dimension six (Zenodo, unrefereed preprint, 2026-09-30). Claims to exclude the Szöllősi family from MUB quadruplets, which bears on the MMVW conjecture mentioned in this paper's introduction (Conjecture 3 and Proposition 4.2 of MatolcsiEtAl2025).
- Gary McConnell, Harry Spencer, Afaq Tahir, Evidence for and against Zauner's MUB Conjecture in ℂ⁶ (arXiv:2103.08703, 2021). Finite-field evidence on bounds for MUBs in dimension six, a context for Theorem 1.2's bound N(6) ≤ 5.
Digests and expositions
- Ying Xie, What the Machine Invented: Mining New Mathematical Instruments from OpenAI's October 2026 Release of AI-Generated Mathematics (arXiv:2610.17090, 2026-10-12). Section 9.3.4 describes the phase cover, tangent refinement and companion Fourier certificates, distinguishes their formalization scopes, and states that the headline computation was not independently rerun for this survey. [AI-generated per source] [family-level]
News and blogs
- Carlos Toledo, OpenAI's math release: what it claims beyond what it checks (audit dated 2026-10-10). Entry F-533 reports exact checks of the single-matrix, cubic-pair and mixed-moment finite certificates at release commit fd4aeeb2, while the separate MUBSix kernel-check entry records that its job stopped before Comparator ran. [cites earlier version]
Elsewhere
- VibeMathed: Problem record
- VibeMathed: Problem record
- Emergent Mind: family 266
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026,
author = {{OpenAI}},
title = {{Exact Fourier certificates for complex Hadamard matrices of order six}},
howpublished = {OpenAI Math Release preprint
\href{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}{OAI:Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026}},
year = {2026}
}From OpenAI's README for this paper:
Verification
Run the two invariant-moment certificates with Python 3.8 or later, using only
the standard library and with assertions enabled (no -O or PYTHONOPTIMIZE):
python3 verification/run.py
The wrapper checks the verifier checksum before running it. Success is a zero exit with the three output lines printed in Appendix A of the PDF. The cubic character counts are separately specified by finite sums and loops in the paper.
Lean formalization
OpenAI provides Lean files for this family: lean/docs/266.md. The Lean statements have not been checked by people against the paper's claims.
Family 266: 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.
Other papers in this family:
- The maximum number of mutually unbiased bases in dimension six · September 24, 2026