An exact MUB triple in dimension six with two Fourier edges

Abstract

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.