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M6_Complete_Proof_Final V-2 AKA The End AKA The Dimension–Six MUB Obstruction AKA A Complete, Closed Proof that M(6) = 3

Pompetzki, Christopher

Published Dec 14, 2025• 0 citations
Publisher

Abstract

The end from here to eternity. We present a single, linear obstruction chain for MUBs in dimension six: a structural lemma(Bridge Lemma) for 6 × 6 complex Hadamard matrices arising as MUB transition matrices,a recognition theorem reducing to two explicit families, and family-specific obstructions. Weinclude explicit invariance and transport lemmas to prevent “hidden equivalence” gaps. TheHaagerup family obstruction is completed by exhibiting an explicit non-scalar unitary commutant.

DOI
10.5281/zenodo.17934029

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The Maximum Number of Mutually Unbiased Bases in Dimension Six - Pompetzki, Chris

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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.

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Unrefereed candidate excluding the specified closed two-circulant Szollosi family from MUB quadruplets in complex dimension six. The Fourier-family result re-proves prior work. The unrestricted dimension-six maximum remains conditional on MMVW Conjecture 2. The package separates an untrusted hull generator from an interval and graph checker; canonical half-open ownership repairs boundary coverage, and input-bound clean replay prevents stale cache acceptance. No end-to-end formal verification, unaffiliated reproduction or external human peer review is claimed. See the replay receipt for exact measured scope. Original prose and data CC0; original code MIT; third-party and review material retain their terms.

•Sep 30, 2026•0 citations
The end from here to eternity. We present a single, linear obstruction chain for MUBs in dimension six: a structural lemma(Bridge Lemma) for 6 × 6 complex Hadamard matrices arising as MUB transition matrices,a recognition theorem reducing to two explicit families, and family-specific obstructions. Weinclude explicit invariance and transport lemmas to prevent “hidden equivalence” gaps. TheHaagerup family obstruction is completed by exhibiting an explicit non-scalar unitary commutant.
Complex Hadamard matrices encode perfectly balanced unitary transformations. Their classification is complete through order five, but order six -- the first dimension in which several continuous families coexist with an isolated solution -- has remained open for decades. Here, we give a complete and exact finite-incidence classification of order-six complex Hadamard matrices up to standard equivalence. We supply the global step missing from Szöllősi's dilation method, which allows us to prove an even stronger version of his conjecture: every complex Hadamard matrix of order six can be recovered algebraically from a suitable, dephased 3×33 \times 33×3 corner defined by four initial phases. We then describe the geometry of the reconstruction from these phases and show that, except for Tao's isolated matrix and a single explicit Karlsson matrix, every class admits a representative obtained by solving one quadratic and one cubic equation in both the horizontal and vertical directions. Our work resolves the classification problem and provides a rigorous framework for further investigating order-six Hadamards, with applications to balanced six-mode interferometers and the study of mutually unbiased bases.
In this research, I constructed an exact algebraic triple of mutually unbiased bases (MUBs) in ℂ⁶ consisting of the computational basis, a member D(ρ) of the Diţă slice from Karlsson's three-parameter family, and an explicit third basis B. Through exact symbolic computation over a degree-96 number field extension, I proved that both remaining transition matrices in the triple belong precisely to the two-parameter Fourier family F(a,b) up to equivalence and transposition, revealing that the Diţă-slice matrix factors directly into a product of two Fourier-type Hadamard matrices. Applying the theorem of Jaming et al., I established that both Fourier edges are non-extendable, proving the triple cannot be extended to any MUB quartet. Additionally, I derived a closed-form expression D(z) for the entire Diţă slice, resolved degeneracies in Karlsson's Möbius parametrization, and computed its 2×2 Hadamard submatrices exactly (51 generically and 75 at z=1), while identifying a second exact Fourier-Fourier triangle at z=1. Finally, using homotopy continuation to track witness systems across seven slice parameters, I recovered 40 third bases that all preserved this Fourier-Fourier pattern to residuals within 5×10⁻¹⁵, providing strong numerical backing for a global structural conjecture along the full Diţă slice.
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.
Unrefereed candidate excluding the specified closed two-circulant Szollosi family from MUB quadruplets in complex dimension six. The Fourier-family result re-proves prior work. The unrestricted dimension-six maximum remains conditional on MMVW Conjecture 2. The package separates an untrusted hull generator from an interval and graph checker; canonical half-open ownership repairs boundary coverage, and input-bound clean replay prevents stale cache acceptance. No end-to-end formal verification, unaffiliated reproduction or external human peer review is claimed. See the replay receipt for exact measured scope. Original prose and data CC0; original code MIT; third-party and review material retain their terms.