A New Non-Affine Family in the Karlsson-Szollosi Intersection

arXiv:2610.33207v1 Announce Type: new Abstract: We show that $K_6^{(2)} \cap X_6^{(2)} = D_6^{(1)} \cup C_6^{(1)}$ up to standard Hadamard equivalence, where $C_6^{(1)}$ is a new non-affine one-parameter family of complex Hadamard matrices. This demonstrates that the intersection of the two-parameter Hadamard families of Karlsson and Sz\"oll\H{o}si is strictly larger than the known affine Di\c{t}\u{a} curve $D_6^{(1)}$. Our new parametrisation of the Sz\"oll\H{o}si family $X_6^{(2)}$ in Karlsson coordinates and our constructive classification of the Karlsson-Sz\"oll\H{o}si intersection shed light on the intersection of block-reducible forms and hypocycloid constraints. This may support the search for mutually unbiased bases in six dimensions.