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Second Rebuttal Bi-Unimodularity is Forced by Orthonormality Why SuperGrok's Second Objection Also Fails

Christopher Pompetzki

Published Dec 30, 2025• 0 citations
Publisher

Abstract

SummarySuperGrok’s second objection claims that the shift-invariance argument in our first rebuttal is flawed because “mutual unbiasedness is not preserved under shifts in general.”This objection is technically correct but entirely irrelevant.The shift argument was an intuitive explanation. The actual proof of bi-unimodularity follows from a different, stronger argument: the orthonormality constraints on bases B and C force the overlap functions to form an orthogonal system, which combined with unimodularity and the Parseval identity, forces flatness of the Fourier spectrum.We present this complete argument.

DOI
10.5281/zenodo.18102666

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Rebuttal - Why SuperGrok's Objection to the MUB Proof is Wrong C.M. Pompetzki (AKA Captain Ahab AKA Shrek)

Christopher Pompetzki

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

•Dec 30, 2025•0 citations

Mutually unbiased bases: common states, completion, and extension obstructions

Hao-Yu Sun

Which pure states give uniform outcomes in several mutually unbiased measurements, and when can they form another measurement basis? We distinguish common unbiased states, their orthonormal completion, and further extension. For a displayed phase-deformed Fourier pair in dimension $2m$, a clock symmetry and skew antiunitary complete every prescribed common unbiased state to an explicit orthonormal basis. In dimension six, an amplitude-profile obstruction proves strong unextendibility of a given mutually unbiased triple when its original third-basis rays are invariant under the ternary clock. We give an exact nonproduct example and an application to a companion whose columns are localized at single Fourier frequencies. For every exact Hadamard matrix in a specified closed entrywise neighborhood of a fixed reference matrix, the coordinate and Hadamard bases have exactly 48 common unbiased rays but no third measurement basis, with a quantitative lower bound on the squared Gram defect of every six-column candidate. We also study auxiliary complex equations for common states, retaining algebraic multiplicities. For a flat unitary of order six with all $3\times3$ minors nonzero, the projective ray scheme is finite; each singular two-minor contributes two reduced isolated boundary points, and the affine coordinate algebra is finite-dimensional over $\mathbb{C}$. In every order, a critical-algebra identity preserves multiplicities; in dimension six it gives finiteness for a fixed phase fibre. Specialization over a discrete valuation ring controls complete reciprocal-orthogonal collections. A nonphysical Tao/Potts collection separates algebraic from physical solutions. These results do not determine the global maximum number of mutually unbiased bases or classify all triples.

quant-pharXiv:2610.12180•Oct 9, 2026•0 citations

Mutually Unbiased Bases in Composite Dimensions What I've Already Done in a More "Proper" Tone Pff Whatever...You Go Ahead And Close An Open Problem of 28+ Years With Not One Single Collaborator And Catch Every Stupid Typo That Some Stooge Will Try to Harp On Don't Make Me Laugh. Objections to Tone Are Not Real Objections. The Literature to Date on This Matter Is Pathetic and Inexcusable. It is a Black Dot On Math History. I Can Write Volumes of Fury On This.

Christopher Pompetzki

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.

•Dec 27, 2025•0 citations

A Rigidity Property of Complete Systems of Mutually Unbiased Bases

Máté Matolcsi, Mihály Weiner

Suppose that for some unit vectors [Formula: see text] in [Formula: see text] we have that for any [Formula: see text] [Formula: see text] is either orthogonal to [Formula: see text] or [Formula: see text] (i.e., [Formula: see text] and [Formula: see text] are unbiased). We prove that if [Formula: see text], then these vectors necessarily form a complete system of mutually unbiased bases, that is, they can be arranged into [Formula: see text] orthonormal bases, all being mutually unbiased with respect to each other.

arXiv:2112.00090•Open Systems & Information Dynamics•Sep 1, 2021•1 citation

The MUB Problem for Composite Dimensions: A Complete Rigorous Treatment Every Step Proven From First Principles Christopher M. Pompetzki

Christopher Pompetzki

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

•Jan 9, 2026•0 citations

M(2p) = 3 for All Primes p ≥ 3 A Complete Proof via Tensor Saturation and Double Flatness Rigidity

Pompetzki, Christopher

AbstractWe prove that the maximum number of mutually unbiased bases in dimension d = 2p, wherep ≥ 3 is prime, equals exactly three: M(2p) = 3. The proof combines tensor product saturationwith a new rigidity theorem: any unimodular sequence on Zp that is flat in both the Fourier basisFp and the Weyl-Heisenberg basis Cp must be a quadratic chirp. This “double flatness” rigidity,proved via character-theoretic arguments on autocorrelation functions, forces all triply-unbiasedstates to be product states, contradicting the existence of a fourth MUB. No restriction to rootsof unity is required. This resolves the MUB problem for infinitely many composite dimensions

•Dec 16, 2025•0 citations
SummarySuperGrok’s second objection claims that the shift-invariance argument in our first rebuttal is flawed because “mutual unbiasedness is not preserved under shifts in general.”This objection is technically correct but entirely irrelevant.The shift argument was an intuitive explanation. The actual proof of bi-unimodularity follows from a different, stronger argument: the orthonormality constraints on bases B and C force the overlap functions to form an orthogonal system, which combined with unimodularity and the Parseval identity, forces flatness of the Fourier spectrum.We present this complete argument.
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
Which pure states give uniform outcomes in several mutually unbiased measurements, and when can they form another measurement basis? We distinguish common unbiased states, their orthonormal completion, and further extension. For a displayed phase-deformed Fourier pair in dimension 2m2m2m, a clock symmetry and skew antiunitary complete every prescribed common unbiased state to an explicit orthonormal basis. In dimension six, an amplitude-profile obstruction proves strong unextendibility of a given mutually unbiased triple when its original third-basis rays are invariant under the ternary clock. We give an exact nonproduct example and an application to a companion whose columns are localized at single Fourier frequencies. For every exact Hadamard matrix in a specified closed entrywise neighborhood of a fixed reference matrix, the coordinate and Hadamard bases have exactly 48 common unbiased rays but no third measurement basis, with a quantitative lower bound on the squared Gram defect of every six-column candidate. We also study auxiliary complex equations for common states, retaining algebraic multiplicities. For a flat unitary of order six with all 3×33\times33×3 minors nonzero, the projective ray scheme is finite; each singular two-minor contributes two reduced isolated boundary points, and the affine coordinate algebra is finite-dimensional over C\mathbb{C}C. In every order, a critical-algebra identity preserves multiplicities; in dimension six it gives finiteness for a fixed phase fibre. Specialization over a discrete valuation ring controls complete reciprocal-orthogonal collections. A nonphysical Tao/Potts collection separates algebraic from physical solutions. These results do not determine the global maximum number of mutually unbiased bases or classify all triples.
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.
Suppose that for some unit vectors [Formula: see text] in [Formula: see text] we have that for any [Formula: see text] [Formula: see text] is either orthogonal to [Formula: see text] or [Formula: see text] (i.e., [Formula: see text] and [Formula: see text] are unbiased). We prove that if [Formula: see text], then these vectors necessarily form a complete system of mutually unbiased bases, that is, they can be arranged into [Formula: see text] orthonormal bases, all being mutually unbiased with respect to each other.
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
AbstractWe prove that the maximum number of mutually unbiased bases in dimension d = 2p, wherep ≥ 3 is prime, equals exactly three: M(2p) = 3. The proof combines tensor product saturationwith a new rigidity theorem: any unimodular sequence on Zp that is flat in both the Fourier basisFp and the Weyl-Heisenberg basis Cp must be a quadratic chirp. This “double flatness” rigidity,proved via character-theoretic arguments on autocorrelation functions, forces all triply-unbiasedstates to be product states, contradicting the existence of a fourth MUB. No restriction to rootsof unity is required. This resolves the MUB problem for infinitely many composite dimensions