A Hilbertian Monograph on the Obstruction to a Fifth Mutually Unbiased Basis in Dimension 3q (Prime q ≥ 5) via Biunimodular Rigidity and the Gauss Phase Law Christopher Michael Mfing Pompetzki (AKA Captain AHAB)
A Hilbertian Monograph on the Obstruction to a Fifth Mutually Unbiased Basis in Dimension 3q (Prime q ≥ 5) via Biunimodular Rigidity and the Gauss Phase LawChristopher Michael Mfing Pompetzki"Towards thee I roll, thou all-destroying but unconquering whale; to the last I grapple withthee; from hell’s heart I stab at thee; for hate’s sake I spit my last breath at thee. Sink all coffinsand all hearses to one common pool! and since neither can be mine, let me then tow to pieces,while still chasing thee, though tied to thee, thou damned whale!" - Captain AhabCall me Ahab not Ishmael. I hunt the damned whale that is MUB. Ahab beckons.December 16, 2025Bibliographic remarksThis monograph is self-contained in the sense that all lemmas explicitly used in the contradictionare proved in full detail. The construction of prime-dimensional Weyl–Heisenberg MUBs and theassociated Gauss-sum calculations are classical (Wootters–Fields). The uniqueness of 3 × 3 complexHadamard matrices is also classical and is included here for completeness.Ahab acknowledges and thanks all the members of the proverbial Pequod who I spam email.
A Hilbertian Monograph on the Obstruction to a Fifth Mutually Unbiased Basis in Dimension 3q (Prime q ≥ 5) via Biunimodular Rigidity and the Gauss Phase LawChristopher Michael Mfing Pompetzki"Towards thee I roll, thou all-destroying but unconquering whale; to the last I grapple withthee; from hell’s heart I stab at thee; for hate’s sake I spit my last breath at thee. Sink all coffinsand all hearses to one common pool! and since neither can be mine, let me then tow to pieces,while still chasing thee, though tied to thee, thou damned whale!" - Captain AhabCall me Ahab not Ishmael. I hunt the damned whale that is MUB. Ahab beckons.December 16, 2025Bibliographic remarksThis monograph is self-contained in the sense that all lemmas explicitly used in the contradictionare proved in full detail. The construction of prime-dimensional Weyl–Heisenberg MUBs and theassociated Gauss-sum calculations are classical (Wootters–Fields). The uniqueness of 3 × 3 complexHadamard matrices is also classical and is included here for completeness.Ahab acknowledges and thanks all the members of the proverbial Pequod who I spam email.
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
The ObjectionSuperGrok claims the following is a “critical logical flaw”:“The claim that |fˆrs(j)| = √d for all j ̸= 0 is justified by ‘unbiasedness to Fourier basis (implied by the MUB structure)’. This is the core flaw. Unbiasedness to the Fourier basis is not implied by general MUB structure. The Fourier basis (the DFT matrix) is unbiased to the computational basis, but in an arbitrary MUB set, the Fourier basis may not be part of the set, and the other bases are not required to be unbiased to it.”This objection reflects a fundamental misunderstanding of the Fourier-analytic structure of mutual unbiasedness. Final VerdictSuperGrok’s “critical logical flaw” is not a flaw. It is a misreading of the proof.The bi-unimodularity lemma stands. The CRT factorization stands. The Z₂ obstruction stands.M (6) = 3."Wir müssen wissen. Wir werden wissen."- David Hilbert of Deutschland, the greatest mathematician of all time
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
We present a detailed computational and algebraic study of Mutually Unbiased Bases (MUBs) in finite-dimensional Hilbert spaces, with a particular focus on dimensions 2, 3, 4, and the challenging case of 6. Starting from the Hadamard-phase parametrization, we derive explicit analytical conditions for mutual unbiasedness in dimension 4, providing a tractable system of trigonometric constraints on the phase parameters. We then explore a tensor-product construction via Pauli operators, highlighting the algebraic and group-theoretical origin of MUBs in two-qubit systems, and demonstrating how these constructions yield a complete set of 5 MUBs in dimension 4. Extending our approach, we investigate the Fourier-family method in dimension 6, where the absence of a prime-power structure imposes strong rigidity constraints and limits the known constructions to sets of 3 MUBs. We provide a systematic computational framework for testing candidate phase vectors, bridging the gap between analytical insight and numerical exploration. Finally, we generalize the discussion to arbitrary prime-power dimensions, emphasizing the role of finite-field structures, Heisenberg-Weyl operators, and discrete symmetries in generating complete sets of MUBs. Our work offers a transparent, line-by-line verification methodology, highlighting both the geometric and algebraic richness of MUBs, and clarifying why certain dimensions resist full analytical constructions. This study serves as a comprehensive resource for researchers seeking both theoretical understanding and practical construction of MUBs in quantum information science.
Abstract. We present a mathematically rigorous unified framework connecting three domains:(1) categorical classification formalism with transfinite stability properties, (2) non-nilpotency obstructionsin Lie algebras of derivations, and (3) structural constraints on mutually unbiased bases(MUBs) in composite dimensions.The central contribution is the Pompetzki Obstruction Framework, which provides precise conditionsunder which pairs of derivations generate non-nilpotent Lie algebras. We prove that infinite-dimensional operator algebras, the relevant obstruction is non-nilpotency of the lower centralseries, not infinite-dimensionality. This correction resolves a technical imprecision in earlierformulations.For MUBs, we prove a conditional obstruction theorem: if mutually unbiased MASA pairssatisfy a spectral non-periodicity condition, then complete MUB families cannot exist. We explicitlycharacterize when this condition holds via the Weyl–Heisenberg Envelope Theorem, connecting finiteLie closure to the existence of nice error bases, which are known to require prime-power dimension.All proofs are self-contained. The satirical “Swamp Formalism” is presented in Appendix Aas a pedagogical toy theory illustrating the axiomatic method, with its model-theoretic propertiesrigorously established. XD
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.