1 · The essentials
The question is how many different ways of describing the same six-dimensional complex space can be equally unfamiliar to one another. Each description uses six perpendicular directions, called a basis. Two such descriptions are mutually unbiased when every direction in one has the same overlap in size with every direction in the other. The main manuscript reports that exactly three mutually unbiased bases can coexist: three exist, but four cannot. Its exclusion of four relies on a complete computer verification under specified arithmetic and compiler conditions. The main result has a Lean formalization, although the supplied summary does not specify its precise coverage. A separate companion manuscript gives exact arithmetic certificates for related six-dimensional results.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine several ways of sorting objects into six drawers: knowing an object's drawer in one sorting scheme gives no preference among the drawers in another. The mathematical question asks how many equally balanced descriptions can coexist, although actual bases are directions in a complex space, not drawers.
Why it matters
The main manuscript identifies its result as resolving Zauner's dimension-six mutually unbiased bases conjecture. It establishes a sharp boundary rather than merely ruling out a very large collection: three bases exist, while four are excluded under the stated computational conditions. The companion work addresses a separate Fourier-vanishing conjecture and provides another certified restriction on mutually unbiased bases.
2 · Going further
Let denote the maximum number of mutually unbiased orthonormal bases in . Mutual unbiasedness means that the magnitudes of the inner products between vectors from different bases are all equal. The main manuscript reports , combining the existence of three bases with an exclusion of four arbitrary complex bases. It describes this as resolving Zauner's dimension-six MUB conjecture.
The upper-bound argument is computer-assisted. According to the abstract, a complete execution of a documented verification pipeline excludes four bases under stated binary64 arithmetic and compiler conditions. The supplied text does not describe the search reduction or the internal steps of that pipeline, so it does not support a more detailed account of the argument. The main result has a Lean formalization, but its exact scope is not specified in the supplied material.
The independent companion manuscript concerns complex Hadamard matrices of order six. It reports the Matolcsi–Ruzsa–Weiner Fourier-vanishing conjecture: outside Tao's cubic equivalence class, every such matrix has a vanishing character sum at every permutation of . It also supplies an exact certificate excluding seven mutually unbiased bases in ; using Weiner's completion theorem, the manuscript derives an upper bound of five. Both companion certificates use only integer and rational arithmetic, and a complete verifier is included. This gives a separate, weaker numerical bound than the main manuscript's sharp bound of three, alongside the Fourier-vanishing result.
Useful notions
- Complex vector spaces
- Inner products and orthonormal bases
- Matrices
- Exact versus floating-point arithmetic
- Computer-assisted verification
What was known before / what is new
Before
The supplied source states that three mutually unbiased bases exist in . It names Zauner's dimension-six MUB conjecture and the Matolcsi–Ruzsa–Weiner Fourier-vanishing conjecture as the problems addressed. It also cites Weiner's completion theorem as the link from excluding seven bases to an upper bound of five.
New
The main manuscript reports a complete computational exclusion of four arbitrary complex bases, yielding the sharp value under its stated arithmetic and compiler conditions. The main result has a Lean formalization. Independently, the companion manuscript reports exact certificates for the stated Fourier-vanishing result and for excluding seven mutually unbiased bases, with a complete verifier included.
Limits and assumptions
The main exclusion is conditional on the stated binary64 arithmetic and compiler conditions, whose details are not supplied here. Although the main result has a Lean formalization, the supplied material does not specify which computational steps or related claims it covers; no Lean coverage of the companion results is stated. The companion Fourier-vanishing claim applies only outside Tao's cubic equivalence class, and its basis-count certificate yields an upper bound of five through Weiner's completion theorem, not the sharp bound of three. The source identifies no further open problems.
Mini glossary
- Orthonormal basis —
- A collection of mutually perpendicular unit vectors that spans the space.
- Mutually unbiased bases —
- Orthonormal bases for which every inner product between vectors from different bases has the same magnitude.
- Upper bound —
- A limit that the maximum possible number cannot exceed.
- Exact arithmetic —
- Arithmetic that keeps values exact, here using integers and rational numbers rather than rounded floating-point values.
- Lean formalization —
- A mathematical result expressed for verification in the Lean proof assistant.
Keywords : Mutually unbiased bases · Dimension six · Certified computation
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Proves , 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.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
The maximum number of mutually unbiased bases in dimension six
24/09/2026We 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
24/09/2026We 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 . 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.