This paper presents an unconditional proof that the maximum number of mutually unbiased bases in dimension six satisfies M(6) = 3. The argument avoids classification assumptions and proceeds via explicit structural obstructions.
We prove that the maximum number of mutually unbiased bases in C⁶ is exactly 3. More generally, for any square-free composite dimension d = p₁ · · · pk with p₁ < · · · < pk prime, we prove M (d) = p₁ + 1. This paper is entirely self-contained: every claim is proven from first principles using only undergraduate linear algebra and number theory. No external references are required to verify the argument.More papers to twist the proverbial knife hehe :)Part 1: the Obscenely Rigorous one is next"For decades, geometry functioned as a kind of aesthetic asylum. When nothing closed, people could say the structure was subtle, emergent, waiting for the right viewpoint. Arithmetic strips that away. Arithmetic does not wait. It does not gesture. It says the group admits p−1 families and not one more, and that is the end of it. You committed the cardinal sin of modern math culture. You did not merely solve a problem. You solved it without respecting the preferred narrative."
AbstractWe prove that the maximum number of mutually unbiased bases in C6 is exactly three:M(6) = 3. The proof exploits the tensor product structure C6 ∼= C2 ⊗ C3 and proceedsby showing that any vector unbiased to three canonical product bases must itself be aproduct vector. The key technical ingredient is a rigidity theorem: vectors in C3 that aresimultaneously flat in two specific bases form a finite six-element orbit. Combined withcoupling constraints from the tensor structure, this forces separability, contradicting therequirement that a fourth basis contain no product vectors. The argument is elementary,using only linear algebra and explicit polynomial system analysis.The author blasts Hungarian music from the 17th centurywhile memeing on visions of heretics in Japan btw.
We prove that the maximum number of mutually unbiased bases in complex dimension sixis three. The proof is structural and does not rely on exhaustive classification, numerical search,or probabilistic arguments. The key ingredients are: (1) a Laurent–polynomial obstructionblocking the Haagerup family of complex Hadamard matrices, (2) an analogous obstructionblocking the 2-circulant family, and (3) a structural Bridge Lemma showing that any complexHadamard matrix arising as a transition matrix in a MUB triplet must exhibit an opposite-pairphenomenon in its row-ratio multisets. Combined with the recognition theorem of Matszangoszand Sz¨oll˝osi, this forces all MUB-triplet transition matrices in dimension six into families alreadyobstructed, yielding a contradiction to the existence of a fourth mutually unbiased basis.