Result 266, Mathematical physics

Exactly three mutually unbiased bases in dimension six

Proves N(6)=3N(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 formalization Proof

The bigger picture

Why it matters

Mutually unbiased bases describe measurement settings where certainty in one means equal probabilities in another. The manuscript claims that six-dimensional complex space permits exactly three such settings, placing a sharp limit on how many can coexist.

What changes?

An orthonormal basis in this space consists of six perpendicular unit vectors spanning the space. Two bases are mutually unbiased when every squared overlap between their vectors equals one sixth. Three such bases were already known to exist. The unreviewed manuscript reports a computer-assisted exclusion of four arbitrary complex bases, establishing a claimed maximum of three. This exclusion depends on a complete execution of its documented verification pipeline under the stated binary64 arithmetic and compiler conditions.

What does that help mathematicians do?

The claimed exclusion would rule out every attempt to extend a mutually unbiased triple to four bases in dimension six, not merely attempts within a particular family of matrices. That turns unsuccessful construction searches into a structural limitation. The companion manuscript also reports an exact certificate excluding seven bases; using Weiner's completion theorem, it obtains the weaker upper bound of five. That certificate uses only integer and rational arithmetic.

Are there practical applications?

The immediate value is foundational for the mathematics of quantum measurements. Mutually unbiased bases encode an exact relationship between measurement settings: a state specified by one basis gives uniform outcome probabilities in another. The claimed bound limits how many such settings six-dimensional models can contain. It does not by itself establish a practical measurement protocol or a technological improvement.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

The maximum number of mutually unbiased bases in dimension six

September 24, 2026 76 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:The-maximum-number-of-mutually-unbiased-bases-in-dimension-six-September-24-2026,
  author = {{OpenAI}},
  title = {{The maximum number of mutually unbiased bases in dimension six}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-maximum-number-of-mutually-unbiased-bases-in-dimension-six-September-24-2026/The-maximum-number-of-mutually-unbiased-bases-in-dimension-six-September-24-2026.pdf}{OAI:The-maximum-number-of-mutually-unbiased-bases-in-dimension-six-September-24-2026}},
  year = {2026}
}

Exact Fourier certificates for complex Hadamard matrices of order six

September 24, 2026 35 pages

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)(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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/266.md.

Exactly three mutually unbiased bases in dimension six

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The paper claims that at most three mutually unbiased orthonormal bases exist in C6\mathbb C^6. The linked formalization proves a weaker family bound: every family in its mutually unbiased bases model has at most five members. It also proves a Fourier character-sum vanishing statement for order-six complex Hadamard matrices not equivalent to the Tao matrix, uniformly over coordinate permutations.

The selected statement does not establish the paper's upper bound of three or its computer-assisted exclusion of four arbitrary bases.

The linked formalization proves a cancellation lemma used in the order-six complex Hadamard analysis. Let HH and its entrywise square both be complex Hadamard matrices, and fix two distinct rows. If the cubes of their six entrywise ratios take only two distinct values, then the sum of the row-ratio terms over either specified cube fiber is zero.

This is a supporting Fourier cancellation statement. The paper's full character-sum vanishing theorem and its mutually unbiased bases bound are outside this selected statement.

Comparator links

Result Comparator statement
Order-six Hadamard Fourier vanishing and a five-basis upper bound MUBSix.lean
Cube-fiber cancellation for order-six Hadamard row ratios HadamardCubeFiber.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.