We resolve the MUB problem for all composite dimensions. For square-free d = p1p2 · · · pk,we prove M(d) = pmin+1 where pmin is the smallest prime factor. The obstruction is arithmetic:the cyclic group Zpmin admits exactly pmin − 1 compatible bi-unimodular families, and thisconstraint propagates through the Chinese Remainder Theorem to bound the full space. Fornon-square-free d, the tensor product construction saturates: M(d) = mini(qi + 1) where theqi are the prime power factors, verified by exhaustive computational search. The famous caseM(6) = 3 falls as a corollary. The problem that has been open for over 28 years is closed. "Aye, aye! and I'll chase him round Good Hope, and round the Horn, and round the Norway Maelstrom, and round perdition's flames before I give him up."- Captain Ahab, MUBy Dick
We resolve the MUB problem for all composite dimensions. For square-free d = p1p2 · · · pk,we prove M(d) = pmin+1 where pmin is the smallest prime factor. The obstruction is arithmetic:the cyclic group Zpmin admits exactly pmin − 1 compatible bi-unimodular families, and thisconstraint propagates through the Chinese Remainder Theorem to bound the full space. Fornon-square-free d, the tensor product construction saturates: M(d) = mini(qi + 1) where theqi are the prime power factors, verified by exhaustive computational search. The famous caseM(6) = 3 falls as a corollary. The problem that has been open for over 28 years is closed. "Aye, aye! and I'll chase him round Good Hope, and round the Horn, and round the Norway Maelstrom, and round perdition's flames before I give him up."- Captain Ahab, MUBy Dick
We provide a fully rigorous, self-contained proof that M (d) = pmin+1 for square-free compos-ite dimensions d, where pmin is the smallest prime factor and M (d) denotes the maximum number of mutually unbiased bases in Cᵈ. Every claim is proven from first principles with no appeals to external results beyond basic linear algebra and number theory. The key technical contributions are: (1) a complete proof that bi-unimodular functions on Zmn (with gcd(m, n) = 1) must factor as tensor products, (2) an explicit classification of bi-unimodular functions on Zp for prime p, and (3) a rigorous counting argument connecting bi-unimodular families to MUB counts. The case M (6) = 3 follows as a direct corollary."Talk not to me of blasphemy, man; I'd strike the sun if it insulted me." - Captain Ahab
Mutually Unbiased Bases in Composite DimensionsChris PompetzkiDecember 2025Fine. Here is the original with the ”proper” tone. Lame. Where’s the fun in that.Whatever. Oh how I much prefer people cite my version with the insufferable tone complete with Moby Dick, Shrek, and Drake references. What else would one expect when one man has done what others haven’t done for what 28+ years now in profound isolation. Though sure, I suppose I’ll do the ”right thing” and post my clean(er) version. Imagine my excitement at the notion of some stooges trying to steal my proof and act like they solved the problem when I actually did. Imagine even claiming it without citing me. Hehe. Oh I hope some do. I do not consent to any posting of this on the pathetic, outdated, obselete arXiv that apparently requires some kind of code to publish. We prove that the maximum number of mutually unbiased bases in dimension d is at most p + 1, where p is the smallest prime factor of d. The proof is three steps: (1) mutual unbiasedness forces bi-unimodularity of overlap functions, (2) bi-unimodularity factors through the Chinese Remainder Theorem to prime factors, (3) the Zp factor admits at most p 1 pairwise compatible bi-unimodular families. For d = 6, this gives M (6) = 3. Oh and by the way, yes I do declare that M (6) = 3 ought to be called the Pompetzki-Ahab-Shrek Theorem or something to that effect. Imagine the audacity of all the people who threw in the towel on M (6) = 3 trying to claim credit for what I, the absolute madman have done. Ha. Only a true psycho would stick with this problem for more than a day tops. Oh I know I ought to have ”proper” toned papers and go get a PhD in pure math apparently. Though then I’d have to do some BS that someone else wants me to do instead of doing what I want to do: brutally advancing mankind’s closure of open problems with all the ”Furor Teutonicus” of zee David Hilbert at least cubed hehe. Ja. Gut. You know vat. I zhought about giving zee Verldt zee fully clean und zee neutered und zee ”proper tone” und zhen I zaid NEIN! Ja. Gut. Vhy. Becauseit makes me smirks. Schadenfreude. That’s why.You know what. I just don't care enough to give it to the world fully clean. I must insist that I coat it with zee Scheisse ja. Zat iz vat is required after all zee Scheisse zat zee stupid have talked about the great German Mathematician, David Hilbert, who is sadly no longer with us. Imagine the extent of his fury at how much of the math world has thumbed their noses at him as they say to Hell with the notion that existence matters in math carry a moral imperative for mathematicians to resolve. I'm too furious. I'm too disgusted. The world didn't deserve David Hilbert. Nor Gauss. Nor Euler. Herr Hilbert said: Wir mussen wissen. Wir werden wissen.Many agreed when he was alive. Then he passed. Then people lost their fear of that Germanic fury. Then M(6) went unclosed for what 28+ years GTFO. There's no excuse for it. I'm furious.
Started From The Bottom (Factor), Now We’re HereThe MUB Problem for Composite Dimensions Christopher PompetzkiLord of the Pequod, Defender of the BottleneckDecember 2025 “There she blows! The fourth MUB!” (Narrator: There was no fourth MUB.) “All visible objects, man, are but as pasteboard masks. Strike through the mask!” We show that the failure of additional MUBs in composite dimensions is not geometric, not analytic, and not quantum. It is a two-element group having a bad day.Specifically: the maximum number of mutually unbiased bases in dimension d is p + 1, where p is the smallest prime factor of d. The proof is constraint propagation through CRT. The obstruction lives in Zp. For d = 6, this gives M (6) = 3, and no amount of entanglement, non-circulant constructions, or optimism can change that. DedicationTo the fourth MUB in dimension six.We searched for you everywhere except where you actually were: nowhere. 1 Statement (This Is Not a Vibe Check)Theorem 1. Let d be composite with smallest prime factor p. Then the maximum number of mutually unbiased bases in Cᵈ isM (d) = p + 1.The lower bound M (d) ≥ p + 1 is achieved by standard constructions. We prove the upper bound.The obstruction to MUBs in composite dimensions is arithmetic and local. It lives in Zp, propagates through CRT, and cannot be avoided by entanglement, exotic constructions, or optimism.Yes, some guy in oh ay North Jowsey has solved a problem that has eluded mathematicians, logicians, scientists, etc. for decades. Yes, I find that hilarious. Will I ever clean up the tone. Hmm. Perhaps. Does it really matter: nope. It's done. I did it first. Feel free to use all my former proof attempts as toilet paper. Oh and that's the thing about proofs. I can make 1,000+ wrong ones. All it takes is one correct one. Ha. Shout out to the real mathematicians who I spam email every time I think I've finished MUBy Dick once and for all.
We resolve the long-standing problem of determining the maximal number of mutually unbiasedbases (MUBs) in arbitrary finite dimension. We prove that maximality alone forcesarithmetic rigidity in transition phases between bases, eliminating all continuous moduli thatobstruct classification in composite dimensions. As a consequence, vanishing-sum constraintsbecome arithmetic rather than geometric, enabling a coprime collision argument governed bythe minimal prime divisor of the dimension. We conclude that for every dimension d ≥ 2, anymaximal family of MUBs in Cd has cardinality 1 + pmin(d), and that this bound is sharp.
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."
Mutually unbiased bases (MUBs) describe quantum measurements for which certainty in one basis gives uniform outcomes in the others. Their maximum number remains unknown in dimensions that are not powers of a prime. We introduce an ansatz that, in many dimensions, allows us to construct more MUBs than have previously been found. Each basis in the ansatz consists of a diagonal phase matrix applied to a fixed unitary, which is a tensor product of Fourier matrices and, optionally, any real Hadamard matrix. This construction yields five MUBs in dimension 12, six in dimensions 48, 96, and 192, seven in dimensions 36 and 108, and ten in dimension 648. These bases can then be extended to exceed the tensor product bound, asymptotically, in every eighteenth dimension. Moreover, we show that Paley's real Hadamard matrix can be used to construct q+1 bases in dimension d=q(q+1) for every prime power q≡3(mod4). This count grows as d and cannot be exceeded by tensor products of smaller sets.